Quasi-continuous vector fields on RCD spaces

Debin, Clément; Gigli, Nicola; Pasqualetto, Enrico

doi:10.48550/arxiv.1903.04302

Cited by 2 publications

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“…Capacity and Hausdorff measures. We briefly recall the notion of capacity and its main properties in this setting, referring to [22] for a detailed discussion on the topic. The capacity of a given set E ⊂ X is defined as…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…• One of the main contributions of [27] was the development of the language of tensor fields defined almost everywhere (with respect to the reference measure) on RCD spaces. In [22] the notion of tensor field defined "2-capacity-almost everywhere" is defined and it is proved that Sobolev vector fields on RCD spaces have a representative in this class.…”

Section: Introductionmentioning

confidence: 99%

“…To handle the first difficulty we mentioned above, we rely on the very recent [22], where a notion of cotangent module with respect to the 2-capacity is introduced and studied. Building upon the fact that the 2-capacity controls the perimeter measure in great generality, we introduce the notion of tangent module over the boundary of a set of finite perimeter (cf.…”

Section: Introductionmentioning

confidence: 99%

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Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Bruè,

Pasqualetto,

Semola

2019

Preprint

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This note is devoted to the study of sets of finite perimeter over RCD(K, N ) metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration-by-parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.

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Section: Preliminaries and Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Bruè,

Pasqualetto,

Semola

2019

Preprint

Self Cite

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“…in a neighborhood of E}. It turns out (see, e.g., [42,Proposition 1.7]) that Cap is a submodular outer measure on X and satisfies m(E) ≤ Cap(E) for every Borel set E ⊂ X.…”

Section: Definition 22 (Bv-functions)mentioning

confidence: 99%

“…In [42] it has been proven that, in situations where continuous functions are dense in W 1,2 (X), there exists a unique map QCR : W 1,2 (X) → L 0 (Cap) that is linear and such that QCR(f ) is (the Cap-a.e. equivalence class of) a function which is quasi continuous and coincides m-a.e.…”

Section: Definition 22 (Bv-functions)mentioning

confidence: 99%

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

Nobili¹,

Violo²

2021

Preprint

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We prove that if M is a closed n-dimensional Riemannian manifold, n ≥ 3, with Ric ≥ n − 1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere S n , then M is isometric to S n . An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the RCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact CD space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGHconverging sequences of RCD spaces and on a Polya-Szego inequality of Euclidean-type in CD spaces.As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the RCD-setting. Contentsq 2. Preliminaries 2.1. Basic notations 2.2. Calculus on metric measure spaces 2.3. CD(K, N ) and RCD(K, N ) spaces 2.4. Polya-Szego inequality 3. Upper bound for α p 3.1. Polya-Szego inequality of Euclidean-type 3.2. Local Sobolev inequality 3.3. Proof of the upper bound 4. Lower bound on α p 4.1. Blow-up analysis of Sobolev constants 4.2. Sharp and rigid Sobolev inequalities under Euclidean volume growth 5. The constant A opt q in metric measure spaces 5.1. Upper bound on A opt q in terms of Ricci bounds 5.2. Lower bound on A opt q in term of the first eigenvalue 5.3. Lower bound on A opt q in term of the diameter 6. Rigidity of A opt q 6.1. Concentration Compactness 6.2. Quantitative linearization 6.3. Proof of the rigidity 7. Almost rigidity of A opt 7.1. Behavior at concentration points 7.2. Continuity of A opt under mGH-convergence7.3. Proof of the almost-rigidity 43 8. Application: The Yamabe equation on RCD(K, N ) spaces 43 8.1. Existence of solutions to the Yamabe equation on compact RCD spaces 44 8.2. Continuity of λ S under mGH-convergence 48 References 50

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Quasi-continuous vector fields on RCD spaces

Cited by 2 publications

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Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

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